Is it possible to go from a narrow and focussed BEng to a more general MSc?

They're within the same field so you should be fine. Obviously your MSc project will have to be in something narrow and specific. My MEng was in aeronautics while my MSc was in ocean and offshore technology.

I can't see there being any issues with that at all. The degrees don't have to align exactly. What stage are you at now? If it's not too late I would suggest doing the MEng over a BEng as it includes an extra funded year which will assist with eventual CEng registration.

My degrees are actually quite similar - they both deal with structures, stress analysis, fluid mechanics, survivability systems, materials and failure mechanisms. I mentioned the MSc project being very focused - mine looked at corrosion on a sunken submarine which I got to go and dive on.

What are you looking at doing after uni?

MEng is an undergraduate degree, and therefore eligible for funding. An MSc isn't, although this may have changed by the time it's relevant to you. As far as I'm aware, most (if not all) engineering degrees have at least the first year in common so you don't have to decide between BEng/MEng right away.

CEng is very useful to have. Some countries/states require professional registration to work there, chartership can be a way of proving equivalent status without having to study in that country. You'll have to check with each country you're looking at.

For banking I would think a quantitative-heavy course like economics or finance would be more useful, and the Diplomatic Service (I assume you're British) just needs a 2:2 in any subject.